Title: Werner's conformally invariant measure for self-avoiding loops on Riemann surfaces


Quantum fields are notoriously complicated because they are distributional in character. In two dimensions scalar quantum fields are "almost functions". Schramm-Loewner evolution (roughly speaking) describes the level sets of conformally invariant fields in two dimensions. I will discuss a recent discovery of Werner (related to the SLE circle of ideas) of a family of measures on self-avoiding loops on Riemann surfaces which is uniquely characterized by a strong form of conformal invariance. This particular result is easy to state, involves relatively little probabilistic sophistication, and is pleasantly reminiscent of the theory of Haar measure. A major open problem is to find an explicit expression for this measure. In one approach (pursued by many authors: Kontsevich and Suhov, Jones et al, Malliavin et al) this involves triangular factorization (or welding), and other forms of factorization, for homeomorphisms of a circle.h